Lotka-Volterra models predator-prey

Use the data of Prob. 5.7 to fit the Lotka-Volterra predator-prey equations (shown below) in order to obtain accurate estimates of the parameters of the model. Modify the Lotka-Volterra equations as recommended in Prob. 5.8, and determine the parameters of your new models. Compare the results of the statistical analysis for each model, and choose the set of equations that gives the best repre.senration of the data. 

Example 13.8 Prey-predator system Lotka-Volterra model The Lotka-Volterra predator and prey model provides one of the earliest analyses of population dynamics. In the model s original form, neither equilibrium point is stable the populations of predator and prey seem to cycle endlessly without settling down quickly. The Lotka-Volterra equations are... 

To overcome these shortcomings, modified predator-prey models were sought for which should improve the most important oversimplifications of the Lotka-Volterra model. In general, the dynamics of two populations, which are coupled by predation, can be generalized as follows... 

Even though the predator-prey model is rather idealized, many kinetic models for real chemical systems are based on it. For example, D.A. Frank-Kamenetsky used the Lotka-Volterra model to explain the processes of higher hydrocarbon oxidation. 

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

First model for oscillating system was proposed by Volterra for prey-predator interactions in biological systems and by Lotka for autocatalytic chemical reactions. Lotka s model can be represented as... 

Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

Fig. 2.13. The random trajectory in the stochastic Lotka-Volterra model, equation (2.2.64). Parameters are a/k = /3/k = 20, the initial values NA = NB = 20. When the trajectory coincides with the NB axis, prey animals A are dying out first and predators second.

In biological dissipative structures, self-oiganization may be related to the attractors in the phase space, which correspond to ordered motions of the involved biological elements (De la Fuenta, 1999). When the system is far from equilibrium, ordering in time or spontaneous rhythmic behavior may occur. The Lotka-Volterra model of the predator-prey interactions is a simple example of the rhythmic behavior. The interactions are described by the following kinetics... 

The Lotka-Volterra model of the predator-prey interactions is a simple example of the rhythmic behavior. The interactions are described by the following kinetics... 

The Lotka-Volterra model is often used to characterize predator-prey interactions. For example, if R is the population of rabbits (which reproduce autocatalytically), G is the amount of grass available for rabbit food (assumed to be constant), L is the population of lynxes that feeds on the rabbits, and D represents dead lynxes, the following equations represent the dynamic behavior of the populations of rabbits and lynxes ... 

This model is popular with many textbook writers because it s simple, but some are beguiled into taking it too seriously. Mathematical biologists dismiss the Lotka—Volterra model because it is not structurally stable, and because real predator-prey cycles typically have a characteristic amplitude. In other words, realistic... 

In the Lotka-Volterra model (3.68)-(3.69), f(P) = P. Model (3.71)-(3.72) is by no means the only possibility. Another popular choice is to assume that the functional response / depends not only on the prey, but it is ratio-dependent, i.e. it depends on the a mount of prey per predator f = f(P/Z), modelling competition for food among predators, that is absent in the previous models. An example is ... 

In these equations, N is the predator and R is the prey species. It is assumed that the dynamics of the prey and predator populations in the absence of the other species is given by exponential growth R = aR or decay N = —bN. Predation is taken into account in the form of a mass action RN term with rates ki and k2- Model (15.1) became famous as the Lotka-Volterra model [11] (some of the work of Volterra was preceded by that of Alfred J. Lotka in a chemical context [12]). 

Going back to D Ancona s problem of inhibited fishing in the Adria, we include the effects of fishing into the Lotka-Volterra model (15.1) by introducing a linear loss term with harvesting rate h for both predator and prey species... 

Fig. 15.3. Pheise portrait of the modified Lotka-Volterra models in the (R, Ai)-phase plane (solid lines). Graphical analysis revolves around plotting the isoclines in the prey-predator phase plane that denote zero-growth of the model predator and prey populations [13] (dashed lines), a) Nonlinear density dependence g R) leads to a decreasing prey isocline and is stabilizing, b) Type-II functional response f R) gives rise to an increasing prey isocline and is destabilizing.

Waltman of the University of Iowa recently pointed out to me in a personal communication that even when Lotka-Volterra concepts are discarded entirely and Monod s model is used for all growth rates, the resulting competition equations for two predators and one prey seem to have limit cycle solutions for certain conditions of operation. Mr. Basil Baltzls has found that use of a so-called multiple saturation model for the predators, which seems to be more appropriate than Monod s model for protozoans at any rate... 

Quite clearly, the growth of a predator population is in some way dependent upon the abundance of its prey. The most frequently cited model of predator-prey dynamics is the set of linked, non-linear differential equations known as the Lotka-Volterra equations (1 ). This model assumes that in the absence of predator, the prey grows exponentially, while in the absence of prey the predator dies exponentially, and that the predator growth rate is directly proportional to the product of the prey... 

Example 13.9 Prey—predator system—Lotka—Volterra model... 

As was mentioned in Subsection 5.6.2, stochastic versions of the Lotka-Volterra model lead to qualitatively different results from the deterministic model. The occurrence of a similar type of results is not too surprising. A simple model for random predator-prey interactions in a varying environment has been studied, staring from generalised Lotka-Volterra equations (De, 1984). The transition probability of extinction is to be determined. The standard procedure is to convert the problem to a Fokker-Planck equation (adopting continuous approximation) and to find an approximation procedure for evaluating the transition probabilities of extinction and of survival. 

Now the question is how to construct the simplest model of a chemical oscillator, in particular, a catalytic oscillator. It is quite easy to include an autocatalytic reaction in the adsorption mechanism, for example A+B—> 2 A. The presence of an autocatalytic reaction is a typical feature of the known Bmsselator and Oregonator models that have been studied since the 1970s. Autocatalytic processes can be compared with biological processes, in which species are able to give birth to similar species. Autocatalytic models resemble the famous Lotka-Volterra equations (Berryman, 1992 Valentinuzzi and Kohen, 2013), also known as the predator-prey or parasite-host equations. 

Vito Volterra (1860-1940), an Italian mathematician, and Alfred J. Lotka (1880-1949), an American mathematical biologist, formulated at about the same time the so-called Lotka-Volterra model of predator-prey population dynamics. The assumptions of this model are ... 

Fig. 3.13 Population trends for predators (dashed line) and prey (solid line) in the Lotka-Volterra model (on-line calculation http /AwUnpei.ac.ru/MCS/Worlaheets/Chem/ChemKin-3-12.xmcd)...

This section is meant to contribute to the old problem of the interaction between two biological species. To be or not to be is the essential question decided by the predator-prey interaction for the members of certain species. The famous Volterra-Lotka model for this problem [4.21,22] has attracted many researchers who have tried to generalize it in many respects [4.1, 6, 8, 9, 23-26]. 

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